Scientific.Functions.Derivatives

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# Module Derivatives

Automatic differentiation for functions of any number of variables up to any order

An instance of the class DerivVar represents the value of a function and the values of its partial derivatives with respect to a list of variables. All common mathematical operations and functions are available for these numbers. There is no restriction on the type of the numbers fed into the code; it works for real and complex numbers as well as for any Python type that implements the necessary operations.

If only first-order derivatives are required, the module FirstDerivatives should be used. It is compatible to this one, but significantly faster.

Example:

     print sin(DerivVar(2))

produces the output:

     (0.909297426826, [-0.416146836547])

The first number is the value of sin(2); the number in the following list is the value of the derivative of sin(x) at x=2, i.e. cos(2).

When there is more than one variable, DerivVar must be called with an integer second argument that specifies the number of the variable.

Example:

       >>>x = DerivVar(7., 0)
       >>>y = DerivVar(42., 1)
       >>>z = DerivVar(pi, 2)
       >>>print (sqrt(pow(x,2)+pow(y,2)+pow(z,2)))

       produces the output

       >>>(42.6950770511, [0.163953328662, 0.98371997197, 0.0735820818365])

The numbers in the list are the partial derivatives with respect to x, y, and z, respectively.

Higher-order derivatives are requested with an optional third argument to DerivVar.

Example:

       >>>x = DerivVar(3., 0, 3)
       >>>y = DerivVar(5., 1, 3)
       >>>print sqrt(x*y)

       produces the output

       >>>(3.87298334621,
       >>>    [0.645497224368, 0.387298334621],
       >>>      [[-0.107582870728, 0.0645497224368],
       >>>        [0.0645497224368, -0.0387298334621]],
       >>>          [[[0.053791435364, -0.0107582870728],
       >>>            [-0.0107582870728, -0.00645497224368]],
       >>>           [[-0.0107582870728, -0.00645497224368],
       >>>            [-0.00645497224368, 0.0116189500386]]])

The individual orders can be extracted by indexing:

       >>>print sqrt(x*y)[0]
       >>>3.87298334621
       >>>print sqrt(x*y)[1]
       >>>[0.645497224368, 0.387298334621]

An n-th order derivative is represented by a nested list of depth n.

When variables with different differentiation orders are mixed, the result has the lower one of the two orders. An exception are zeroth-order variables, which are treated as constants.

Caution: Higher-order derivatives are implemented by recursively using DerivVars to represent derivatives. This makes the code very slow for high orders.

Note: It doesn't make sense to use multiple DerivVar objects with different values for the same variable index in one calculation, but there is no check for this. I.e.:

       >>>print DerivVar(3, 0)+DerivVar(5, 0)

       produces

       >>>(8, [2])

but this result is meaningless.

Classes
| DerivVar Numerical variable with automatic derivatives of arbitrary order

Functions
Scientific.Geometry.Vector |
DerivVector(x, y, z, index=0, order=1) Returns: a vector whose components are DerivVar objects |
`bool` |
isDerivVar(x) Returns: True if x is a DerivVar object, False otherwise |

Function Details

### DerivVector(x, y, z, index=0, order=1) |
Parameters: `x` (number) - x component of the vector `y` (number) - y component of the vector `z` (number) - z component of the vector `index` (`int`) - the DerivVar index for the x component. The y and z components receive consecutive indices. `order` (`int`) - the derivative order Returns: Scientific.Geometry.Vector a vector whose components are DerivVar objects

### isDerivVar(x) |
Parameters: `x` - an arbitrary object Returns: `bool` True if x is a DerivVar object, False otherwise